The imperfection data bank and its applications

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Keywords: imperfection data bank, lower bound, buckling

Copyright c_{2009 by J. de Vries}

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, including photocopying, recording or otherwise, without the prior written permission of the author J. de Vries, Nederlandse Defensie Academie, Faculteit Militaire Wetenschappen, P.O. Box 10000, 1780 CA Den Helder, The Netherlands.

Cover design by Peter J. de Vries, Multimedia NLDA/KIM Printed in The Netherlands by Giethoorn ten Brink

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The Imperfection Data Bank and its

Applications

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College van Promoties,

in het openbaar te verdedigen op maandag 11 mei 2009 om 15.00 uur

door Jan DE VRIES

ingenieur luchtvaart en ruimtevaart

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Dit proefschrift is goedgekeurd door de promotoren: Prof.dr. Z. G¨urdal

Prof.dr.ir. A. de Boer

Samenstelling van de promotiecommissie: Rector Magnificus

Prof.dr. Z. G¨urdal Prof.dr.ir. A. de Boer Prof.dr. J. Arbocz Prof.dr.ir. A. Verbraeck Prof.dr.ir. M.A. Guti´errez Dr. M.W. Hilburger Prof.dr. A. Rothwell

voorzitter

Technische Universiteit Delft, promotor Universiteit Twente, promotor

Technische Universiteit Delft Technische Universiteit Delft Technische Universiteit Delft NASA Langley, U.S.A., adviseur

Technische Universiteit Delft, reservelid

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Nomenclature

A11 Extensional stiffness of anisotropic shell in axial direction

A22 Extensional stiffness of anisotropic shell in circumferential direction Akℓ Fourier coefficient

Aℓ Fourier coefficient (ring) Ar Ring area

As Stringer area Bkℓ Fourier coefficient Bℓ Fourier coefficient (ring) C Extensional stiffness constant c1 Stringer width

c2 Ring width

Ckℓ Fourier coefficient

D Bending stiffness constant d1 Stringer height

D11 Bending stiffness of anisotropic shell in axial direction d2 Ring height

D22 Bending stiffness of anisotropic shell in circumferential direction Dkℓ Fourier coefficient

dr Ring spacing ds Stringer spacing

Dx Bending stiffness of shell plus smeared out stiffeners, in axial direction

Dy Bending stiffness of shell plus smeared out stiffeners, in circumferential direction ix

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x CONTENTS

E Modulus of elasticity er Eccentricity of the ring es Eccentricity of the stringer

Ex Extensional stiffness of shell plus smeared out stiffeners, in axial direction

Ey Extensional stiffness of shell plus smeared out stiffeners, in circumferential direc-tion

F.S. Factor of safety

H Height of conical shell

h Optimal bin width in histogram

Ip Polar moment of inertia of the core inside the shell in the Stonivoks and Univimp test setup

Ir, I02 Moment of inertia of the rings Is, I01 Moment of inertia of the stringers k Wave number in axial direction

ℓ Wave number in circumferential direction

L Length of cylindrical shell or slant length of conical shell lr Length of the rod used in ECCS handbook

M Maximum wave number in axial direction m Shape parameter in Weibull distribution

N Maximum wave number in circumferential direction n Number of observations

Pa Allowable applied load

Pani Classical buckling load for anisotropic shells Pc Lowest buckling load of the perfect structure Pcl Classical buckling load

Pexp Experimental buckling load

Pstf Classical buckling load for orthotropic shells Pγ Lower bound buckling load

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CONTENTS xi ¯

Qij Stiffness parameter of a layer

Qij Reduced stiffness parameter of a layer R Radius of the shell

S Minimum value in least squares method t Wall thickness of the shell

tγs,n−1 Student’st variable for a confidence level of 100 × γs% and a sample sizen

t+ _{Adjusted wall thickness for anisotropic shells} t∗

Wall thickness, smeared out stiffeners included tu Unified thickness

¯

w Imperfection, positive outward ¯

wA Imperfection at roller A ¯

wB Imperfection at roller B ¯

wC Imperfection at position of displacement transducer X1 Offset inX direction

Y1 Offset inY direction ¯

Z Modified Batdorf parameter ¯Z = L2_/Rt zi Layer coordinate

α Angle between roller and transducer

α Threshold parameter in lognormal or Weibull distribution αs Significance level

αc Cone angle

β Scale parameter in Weibull distribution γ Knock-down factor in lower bound formula γs Confidence level

δAB Distance between the two rollers A and B ε1 Angular offset

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xii CONTENTS

η01 Geometric parameter of the stringers η02 Geometric parameter of the rings λ Normalized buckling load λa Improved knock-down factor λm

Ckℓ Critical (lowest) eigenvalue of the linearized stability equations using membrane prebuckling

λmnτ Critical (lowest) eigenvalue of the linearized stability equations using membrane prebuckling of the anisotropic shell

µ Mean of a distribution

µ1 Geometric parameter of the stringers µ2 Geometric parameter of the rings µ Sample mean

µL Lower bound of the mean of a distribution ν, νij Poisson’s ratio

ˆ ξ Imperfection parameter, ˆξ =qA2 kℓ+ Bkℓ2 or ˆξ = q C2 kℓ+ Dkℓ2 ρ Normalized buckling load, orthotropic shells

σ Standard deviation of a distribution σ Sample standard deviation

σc Critical buckling stress σcl Classical buckling stress

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Abstract

The main objective of this thesis is to describe the creation of an imperfection data bank and tools to process the data. Imperfections are irregularities of the shape of a thin-walled shell, such as those used for rocket structures or silos. Knowing the imperfections is very important as thin-walled shells are very sensitive to imperfections. Even a small deviation with respect to the perfect shell shape reduces the buckling load significantly. Rocket shells have been designed and built for many years. A typical design procedure of a shell is:

a. Define vehicle performance requirements. b. Lay-out preliminary dimensions.

c. Determine loads and environments.

d. Select structural concept (e.g., wall construction and material).

e. Select design and safety factors, including shell buckling knock-down factor that accounts for the degrading affect of the geometric imperfections on the buckling load.

Point of investigation in this thesis is the question if the knock-down factor can be opti-mized. The current factor is too conservative for most of the shells. This is caused by the fact that the knock-down factor, as can be found in the NASA report SP-8007 [1], is based on old testdata. Shells have been tested for some decades and in many cases both the buckling load and the imperfections have been measured. In Delft for instance many reports including test data have been written, also in many other places such reports exist. It is clear a lot of data exists, however this data is not readily available. It is stored in different places, in different formats, sometimes even on ancient storage devices which are becoming increasingly difficult if not impossible to read using modern devices.

As part of this research an imperfection data bank has been created in which most of the available measured data have been stored. These data had to be collected, analyzed, and very often rewritten into the standard format used in the data bank. An interface has been written which enables users to have user friendly access to the data bank. This interface has been written as a web application, thus making it accessible via the Internet. The data have also been protected against deletions or modifications, by ensuring the interface allows for read-only access.

The interface not only facilitates retrieving measured data from the data bank, it also has many features to analyze sets of data. For example, lower bound plots can be gener-ated for all or user selected sets of tests. Furthermore, a lot of effort has been put in the

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xiv CONTENTS

analysis of the Fourier coefficients used in the representation of the imperfection fields. Using the imperfection data bank allows the reproduction of existing reports of test results using only a few mouse clicks.

It has also been shown that similar shells have similar imperfections. It would be very interesting which imperfection are caused by a certain production process. The term manufacturing signature was introduced by Starnes [2]: every production process will yield a certain type of imperfections. In this thesis it has been shown that the imperfections are not related to where a shell was produced. Using the state of the art technology to produce new shells the usage of the common design curves. i.e. the lower bound curves would yield a very conservative, too heavy, design. Thus each of these manufacturing processes deserves its own lower bound. These improved lower bounds were not derived, however the usage of the imperfection data bank filled with sufficient data could very well assist in this. This is also one of the recommendations, to perform many tests of new shells and store them into the data bank.

The test equipment used for imperfection measurements and available at the Uni-versity of Technology in Delft will be described. The smallest installation Stonivoks is capable of automatically measure the imperfections of small beer cans. The medium test setup Univimp is configured to measure shells with diameter 240, 360, and 480 mm. Other shell diameters are possible, however this requires production of new end rings. The largest test facility Amivas is used to measure the imperfections of full scale rocket interstages or satellites. This equipment is flexible in this sense that it only requires minor modifications to measure a different type of shell. Amivas has been used to measure the imperfections of the VEGA interstage 1/2.

In the statistical analysis on sets of shells a distinction is made between input and output statistics. Starting with the latter, it is possible to look at average and standard deviation of buckling loads. Using input statistics it is possible to calculate these parame-ters on all of the Fourier coefficients separately. Using the most significant Fourier terms to generate an average imperfection field the buckling behaviour of a shell is calculated. Hilburger et al. [2] proposed an approach to use the average imperfection plus standard deviation to predict the lower bound of a composite shells, using some simplifications. It has been shown that this theory cannot be used for isotropic shells.

As a general recommendation it should be noted that the research on the buckling behaviour of thin walled shells has to continue. The imperfection data bank can be a tool to be used together with the general shell design codes. As such it has to be updated with test results of both laboratory models as full scale models and real space worthy rockets. Especially the composite shells are still a minority in the data bank and therefore need attention. As a final remark: the data bank is a living environment, it should keep growing. Keeping it alive will be the best thing for letting it be used by the structural designers.

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Samenvatting

Het hoofddoel van dit proefschrift is het maken van een imperfectie databank en gereed-schap om de data te bewerken. Onder imperfectie wordt verstaan een vormonzuiverheid van een dunwandige schaal zoals bijvoorbeeld een raketconstructie of een graansilo. Het is zeer belangrijk dat men weet hoe die imperfecties eruit zien omdat dunwandige schalen hier heel gevoelig voor zijn. Een kleine afwijking ten opzichte van een perfecte schaal zal de kniklast al significant laten dalen. Al vele jaren worden er al raketten gebouwd zonder de imperfectie databank. Het ontwerpproces van een raket ziet er als volgt uit:

a. Defini¨eren van de vereiste prestaties. b. Opzet van de voorlopige dimensies.

c. Bepalen van de belastingen en randvoorwaarden.

d. Selectie van een concept voor de constructie (zoals de huidconstructie en het mate-riaal).

d. Kies ontwerp en veiligheidsfactoren, inclusief de knock-down factor voor het knik-ken van de schaal die het verlagen van de kniklast door de geometrische imperfec-ties in rekening brengt.

In dit proefschrift wordt gekeken of de knock-down factor aangepast kan worden. Het probleem is namelijk dat deze factor in het algemeen veel te conservatief is. De reden hier voor is dat de knock-down factor zoals bijvoorbeeld in het NASA rapport SP-8007 [1] gebruikt wordt, gebaseerd is op heel oude meetdata.

Er worden al decennia lang testen op schalen uitgevoerd. Naast de meting van de kniklast zijn ook de imperfecties van de schalen gemeten. In Delft is een hele serie rap-porten met testdata geschreven, en ook op andere plaatsen is dit gedaan. Het is duidelijk dat er veel data bestaat, echter deze data is niet direct toegankelijk. Het is opgeslagen op verschillende plaatsen, in verschillende formats. Soms ook nog op antieke opslagmedia die moeilijk of soms helemaal niet leesbaar zijn.

In dit werk is een imperfectie-databank gebouwd waar meetdata in is opgeslagen. De data is verzameld, geanalyseerd, en indien nodig omgeschreven naar het format gebruikt in de imperfectie-databank. Er is een interface geschreven die het voor de gebruikers gemakkelijk maakt om toegang tot de databank te krijgen. Deze interface is geschreven als een webapplicatie zodat de databank toegankelijk is via internet. De databank is be-schermd tegen onverhoopte modificaties of verwijderingen van data omdat de interface alleen een leesmogelijkheid heeft.

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xvi CONTENTS

De interface maakt het niet alleen gemakkelijk om data uit de databank te halen, er zijn ook een aantal programmas ingebouwd om data te analyseren. Er kunnen zgn. lower

bound plots van de kniklasten van alle of een geselecteerd aantal schalen geplot worden.

Bovendien is er veel aandacht besteed aan de Fourier coeffici ¨enten die gebruikt worden in de beschrijving van de imperfectie velden. Door gebruik te maken van de imperfectie-databank kunnen bestaande rapporten met test resultaten eenvoudig met enkele klikken met de muis opnieuw gemaakt worden.

Gelijksoortige schalen hebben gelijksoortige imperfecties. Men zou graag de vorm van de imperfecties willen weten die inherent zijn aan een bepaald productieproces. De term manufacturing signature werd door Starnes [2] geintroduceerd: elk productie proces zal een bepaald type imperfecties veroorzaken. In dit proefschrift wordt aangetoond dat deze imperfecties niet gerelateerd zijn aan wie de schaal geproduceerd heeft of waar dat gebeurd is. Als de nieuwste technieken gebruikt worden om de schalen te produceren zal het gebruik van de gebruikelijke ontwerpkrommes, dus de lower bound krommes, een zeer conservatief ontwerp opleveren, en daarmee een te zwaar ontwerp. Voor elk productie proces is daarom een specifieke lower bound een vereiste. Deze krommes zijn hier niet afgeleid, de imperfectie-databank kan echter wel gebruikt worden als hulp bij het opstellen er van. Een van de aanbevelingen is dan ook om nog veel meer test gegevens te verzamelen en nieuwe tests uit te voeren en deze in de databank te zetten.

De test apparatuur voor imperfectie metingen op de Technische Universiteit in Delft is beschreven. De kleinste installatie is Stonivoks. Dit apparaat kan volledig automatisch de imperfecties van bierblikjes opmeten. Het middelgrote apparaat Univimp is zodanig geconfigureerd dat het schalen met een diameter van240, 360 en 480 [mm] kan opmeten. Andere diameters zijn mogelijk, maar vereisen de productie van nieuwe eindringen met aangepaste diameter. De grootste testopstelling betreft Amivas. Deze kan gebruikt wor-den om de imperfecties van echte raketsecties of satellieten op te meten. Dit apparaat is heel flexibel: er zijn slechts kleine modificaties nodig voor het meten van verschillende groottes van schalen. Met Amivas zijn de imperfecties gemeten van de VEGA tussensec-tie 1/2.

In de statistische analyse van verzamelingen van schalen wordt een onderscheid ge-maakt tussen invoer en uitvoer statistiek. Om met de laatste te beginnen, het is bijvoor-beeld mogelijk te kijken naar de gemiddelde waarde en de standaard deviatie van de kniklasten. Met invoer statistiek is het mogelijk deze parameters te berekenen van alle Fourier coeffici¨enten apart. Gebruik makend van de grootste Fourier coeffici¨enten wordt een gemiddeld imperfectie veld berekend. Van een schaal met dit laatste imperfectie veld wordt vervolgens de kniklast berekend. Hilburger et al. [2] hebben een benadering voor-gesteld om de gemiddelde imperfectie plus standaard deviatie te gebruiken om de lower

bound te voorspellen, waarbij enkele vereenvoudigingen zijn gebruikt. In dit proefschrift

wordt aangetoond dat deze theorie niet geldt voor isotrope schalen.

Als algehele aanbeveling kan gesteld worden dat het onderzoek naar het knikgedrag van dunwandige schalen gecontinueerd dient te worden. De imperfectie databank kan als een gereedschap samen met de algemene schaal ontwerpcodes gebruikt worden. De databank moet daarom steeds up to date gehouden worden met zowel de testgegevens van laboratoriummodellen en modellen op volledig schaal, naast gegevens van echte ge-certificeerde raketten. De composietschalen zijn momenteel nog in de minderheid in de

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CONTENTS xvii

databank en vereisen derhalve speciale aandacht. Tot slot: de databank is een levende omgeving, het zal moeten blijven groeien. Hier zijn de constructie ontwerpers het meest bij gebaat.

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Chapter 1 Introduction

The beer can was denied its original purpose in life. Before it got to the filling station in the beer plant, it got removed from the machine to be of use in the investigation of imperfection sensitivity of thin-walled shells. As it found out what was going to happen, the beer can reconsidered what to do. It could not taste the beer it had waited for for so long. However, this was not an unrealistic thought. Serving as a container for some liquid, whilst not being able to drink, and waiting for some person to come along and empty you and then get thrown out of the window if you were unlucky, or get recycled if you weren’t. No, one had to look for new opportunities. What was this imperfection sensitivity all about? Thin-walled shells, that is me, it thought. Am I alone in this world or are there more like me? Yes, I know lots of fellow beer cans. Even some vague far away families who prefer cola or orange juice even. But they are all small like me. The can then found out that there are huge shells, dinosaur tall compared to him, but not extinct. They did not contain stuff like beer, or coke, but very interesting sounding stuff like LOX or LH2. The can did not know what kind of stuff this was, but realized this: these big brothers were about to fly to the moon, to Mars or even maybe out of the solar system. No short life time, no low mile coverage, no, just your ordinary Saturday evening getting sold, getting drunk and getting thrown away. These guys really went somewhere. Now this was something to think about. The little can thought that even though he could not fly into space, it would also mean a lot to him if he could in some way help his big friends to safely fly into the sky.

1.1 The shell design procedure

Thin-walled stiffened or unstiffened, metallic or composite shells are widely used struc-tural elements in aeronautical and space applications. These structures are often highly sensitive to initial geometric imperfections and therefore have buckling loads much lower than those computed for perfect structures. In this thesis the emphasis lies on geomet-ric imperfections of thin-walled shells. Other types of imperfections also exist such as the thickness variation of shells, which is found for composite shells. The layers in the composite shells can have overlaps locally resulting in a larger thickness. The geometric

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2 Introduction

imperfections are also known as mid-surface imperfections. These mid-surface imperfec-tions are sometimes referred to as the traditional imperfecimperfec-tions of a shell [3]. Another important imperfection is the so-called boundary imperfection: if the ends of the shell show some irregularities, or if the end-rings in which the shells are mounted are not com-pletely flat, the load on the shell is not a constant line load. The boundary imperfection and the thickness variation are non-traditional imperfections.

When a structural engineer designs a new light-weight structure like a thin-walled shell he is used to follow the guidelines as in the NASA report SP8007 [1]. A typical design procedure used for the layout of such structures can be summarized as follows:

a. Define vehicle performance requirements. b. Lay-out preliminary dimensions.

c. Determine loads and environments.

d. Select structural concept (e.g., wall construction and material).

e. Select design and safety factors, including shell buckling knock-down factor that accounts for the degrading affect of the geometric imperfections on the buckling load.

In this lower bound design philosophy the following buckling formula is used: Pa≤

γ

F.S. Pc (1.1)

where Pa = allowable applied load; Pc = lowest buckling load of the perfect structure; γ = ”knock-down” factor; and F.S. = factor of safety.

The design requirements specify that the loads should not exceed the limit loadγPc, but a certain amount of reserve strength against complete structural failure is necessary. In aerospace industry the allowable or ultimate loads are equal to the limit loads divided by a factor of safety. In general the factor of safety is 1.5. Notice also that the ultimate loads should be carried by the structure without failure.

There is another way one can look at safety factors. Depending on who will be the users of a structure the safety factor could be set to a different value.

Suppose one introduces three new kinds of safety factors, i.e.F.S.C,F.S.LandF.S.T. They account for the following:

F.S.C where C stands for ’Chiel’. Chiel is the clever person, very accurate worker,

precise. If he builds something it is perfect. This parameter is chosen as F.S.C = 0.97

since structure will carry more load than one would normally expect because of the fine art work.

F.S.L where L stands for ’Loes’. She will look at a structure and decide it is nice

but it needs some colours, maybe we put stickers on it as well, hereby introducing extra weight and eccentricities. The parameter is chosen asF.S.L = 1.1.

F.S.T whereT stands for ’Tom’. This guy has a destructive principle. His philosophy

is that engineers probably put large safety factors on structures. So if a structure could withstand a certain load, he would have no problem going much above this load. This parameter is chosen asF.S.T = 1.4.

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1.2 Why are imperfections important for the design of shells? 3

This thesis will not suggest a new setup of the usage and magnitude of the factors of safety, but will introduce new possibilities of increasing the limit load by improving the shell buckling knock-down factor.

Equation (1.1) provides a good lower bound for most test data. The shell, if so de-signed, will be a safe design: it will be able to bear the absolute maximum load without failing. It will probably be a very conservative design also. In most cases the initial im-perfections in a shell are unknown. Therefore those imim-perfections cannot be taken into account when solving the stability problem using an analysis code. One could of course measure those initial imperfections for each shell, this is however a costly matter. Be-sides that, in the design process, one will consider several concepts of a shell, which will exist only on paper and the actual imperfection may not be known. It would be conve-nient if one had some idea on what the imperfections would look like. The imperfections might appear to have a random character, however, it will be shown here that they can be linked to manufacturing processes. Fortunately, those individuals and research institutes involved in shell research often collect information about imperfections.

For example, let us compare the measured imperfections of two shells, the so-called AS 2 from Caltech and KR1 from Technion. The first shell, AS 2 was measured by Singer, Arbocz and Babcock in 1969 in the California Institute of Technology [4, 5]. The second one, KR1 was measured by Abramovich, Ronith, Grunwald and Singer in 1977 in Israel at the Technion Israel Institute of Technology [6]. Both shells were manufactured by different people, in different places. The manufacturing process was the same. Plots of the initial imperfections are reproduced in Figure 1.1. At first sight the imperfections of both shells look rather different. If one describes the imperfections using Fourier series, as will be explained further in Chapter 4, for each of these shells a number of Fourier coefficients can be calculated. In Figure 1.2 the circumferential variation of the half-wave cosine Fourier representation is plotted for both shells. Comparing both shells by looking at the Fourier coefficients in this figure, it seems that the shells from Caltech have been manufactured more accurately since the coefficients are smaller than those of the Technion shell. The sizes of the Fourier coefficients corresponding to the circumferential wave number where the axial half wave numberk = 0 show a similar distribution, albeit differing a factor of two.

The imperfection data of shells manufactured using the same fabrication process can be used to create reliability functions. To do this a stochastic method like the Monte Carlo Method or the First Order Second Moment method may be used. For a given reliability an analytical knock-down factor λa can be determined [3]. This λa will replace γ, the known very conservative knock-down value from NASA SP8007. The parameterλawill be called an improved knock-down factor.

1.2 Why are imperfections important for the design of

shells?

The design of cylindrical shells involves participation of individuals from different seg-ments of the engineering world. In the first place there will be a customer who is request-ing a particular type of shell. Then the structural engineer will come up with a design,

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4 Introduction 0 1 2 3 0 90 180 270 360 θ = y/R w/t L 0.5L x Caltech AS 2 0 1 2 3 0 90 180 270 360 θ = y/R w/t L 0.5L x Technion KR1

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1.2 Why are imperfections important for the design of shells? 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 20 25 im p er fe ct io n ˆ ξ= q A 2 kℓ + B 2 kℓ

circumferential wave numberℓ

k = 0 k = 1 k = 2 k = 3 Caltech AS 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 im p er fe ct io n ˆ ξ= q A 2 kℓ + B 2 kℓ

circumferential wave numberℓ

k = 0 k = 1 k = 2 k = 3

Technion KR1

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6 Introduction

which in turn is built in the factory by the production people. The final product is returned to the customer. This is the design process in a nutshell.

It is a well known fact that cylindrical shells are sensitive to imperfections, reducing their load carrying capability substantially [7, 8, 9]. In the design process of a shell the imperfections will not be known as the shell is still to be produced. Knowing the exact imperfections of a shell would be the best solution for predicting the buckling load and buckling mode of the shell. If one does not know the imperfections, assumptions will have to be made. Of course after the shell has been built up, it should be verified if the assumptions were acceptable. Measuring the imperfections of each shell takes time, and money. Even more if the assumptions were optimistic and one should start all over.

To take into account the influence of these imperfections, it is common practice in in-dustry to calculate the eigenmodes associated with the lowest eigenvalues of the shell [10]. If the imperfections in the structure resemble the eigenmodes or a combinations of these modes, the reduction in the buckling load will be the largest [11]. To calculate the buck-ling behaviour of a shell where the imperfections are composed of a set of eigenmodes corresponding to the lowest eigenvalues, is a relatively cheap operation compared to the use of the real imperfections obtained using expensive testing. Furthermore, if the eigen-modes will be used as the assumed imperfection shape, it also needs to be decided what magnitude to choose. On the other hand, if the imperfections do not resemble the eigen-modes, the calculated buckling load will be lower than the real one, yielding a conserva-tive and therefore heavy design.

Suppose the imperfections could be related to production methods, to the quality of the processes. Choosing a certain production process, the design engineer then knows what the imperfections will look like. As an example one can think of an interstage of a rocket. The interstage is built up of say 6 curved panels, jointed by offset lap splices. Measuring the imperfections of this shell will definitely show the curved panels because of the appearance of 6 circumferential waves.

1.3 Building an imperfection data bank

In the last decades a lot of imperfection measurements on thin-walled shells have been performed. In the beginning of the 20th century only the buckling load and buckling modes were measured in tests [12, 13, 14, 15, 16, 17]. The data is only available as published papers containing tables with test data and photographs showing the buckled shells. In the sixties Arbocz [18] started to measure the imperfections also. Along with the published papers presenting the results, the data is also digitally stored. In the following years, in several countries, researchers measured imperfections and buckling loads on several types of shells [6, 19, 20, 21, 22, 23, 24, 25]. The first initiative to the data bank was started in 1979 by Arbocz and Abramovich with the report ’The Initial Imperfection Data Bank at the Delft University of Technology Part I’ [5], followed by Parts II - VI [23, 24, 26, 27, 28]. These reports contain the data of tests carried out at Caltech in the sixties of the last century, and tests on ARIANE interstages produced by Fokker. There are much more experimental data consisting of buckling load and imperfection data of shells available, but these have been stored in many companies and universities in different

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1.4 Layout of the thesis 7

countries. Thus, it is hard to get an overview of all data, or to have access to them. It would be very convenient if all data would be accessible to all designers. Unfortunately, this is not that easy because companies may have spent a lot of money on the tests, or data might have restricted access because of security issues (defence technology!). After one has succeeded in getting a set of test data, one will notice that different institutions use different formats to store their data. Different units have been used: in Europe the SI units are very common, in the United States many companies are still using Imperial Units.

In order to improve the knock-down factor in the lower bound formula for the buckling load the influence of imperfections is subject of several research programs [2, 3, 18, 29, 30, 31]. This has lead to the following research questions:

• Is it possible to collect all available data of thin-walled shells and make them inter-actively accessible to shell designers and researchers?

• Can a relation be found between the imperfections and the manufacturing process of a shell?

• Can statistical analysis using the tools of the interface of the imperfection data bank help in the design of the shells?

To answer the first research question, published papers containing test results of shells need to be collected. Next, datasets containing experimental data should be gathered from all over the world. Next, the data needs to be digitized if needed and stored in a computer system. An obvious choice for the latter is the creation of a data bank. It is the primary purpose of this thesis to develop an imperfection data bank to store measured imperfections to be made available to a world wide community of engineers. Along with the data bank, tools to interrogate the data have been developed so that designers will be more flexible in the design of new reliable shells similar to the ones included in the data bank. Access to the data for shell designers and researchers working in different countries can be made possible by connecting the data bank to the internet.

1.4 Layout of the thesis

One of the reasons that the stability of axially loaded thin-walled shells has been the subject of research for so many years, is the large discrepancy between the theoretically buckling load and the experimentally found value, both stored in the imperfection data bank. A workaround of this problem was the introduction of the lowerbound [1]. The traditional lower bound design philosophy is described in Chapter 2. Plots created by the imperfection data bank are shown containing a lower bound curve and a collection of experimental buckling data. In order to store the imperfection measurements in a data bank, one has to first come up with measurement procedures that will accurately produce the data desired. As part of this thesis a procedure was developed and imperfections have been measured using different measurement equipment. The test equipment available at the Faculty of Aerospace Engineering of the University of Technology Delft is described in Chapter 3. This chapter starts with a short overview of the history of imperfection data measurement. A generic test procedure for the imperfection measurement is described.

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8 Introduction

Also the measuring of the new VEGA launcher vehicle currently under development by ESA is described including the processing of the raw data. Once the imperfections are measured, the data has to be presented to the users of the data bank in a convenient amd meaningful fashion. Several ways to represent imperfections have been described in Chapter 4. As an example the Fourier coefficients of the imperfection of the VEGA interstage are determined. Subsequently these imperfection are compared with the ARI-ANE interstage data measured some years earliers. Analyzing all the available data, and executing a test has made it clear which data needs to be stored in the data bank. This has lead to the design of the data bank in Chapter 5. The design of the imperfection data bank is described, starting with all its requirements. Also some technical background is given. The usage of the interface is demonstrated by showing how to retrieve data from the data bank of Arboczs favorite A-shell. This part of the chapter could be a good starting point for an engineer who is interested in using the data bank. The final chapters deal with the application of the imperfection data bank for statistical analysis. More precisely, the statistical tools are discussed in Chapter 6. In Chapter 7 these tools have been used on the research of the buckling behaviour of a shell with averaged imperfection. In the last chapter some general conclusions and recommendations are presented.

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Chapter 2 Lower Bound Design Philosophy

”I do not want to know, there-fore I will not measure” [32]

In Chapter 1 the necessity of measuring the imperfections of thin-walled shells was dis-cussed. Basically the lower bound theory is often used if there is no imperfection data available, and one lacks time and/or money to obtain them. In this chapter the lower bound theory will be explained. A difference is made between isotropic, orthotropic and anisotropic shells. Isotropic shells have been manufactured from metal plates with material properties which do not depend on the direction. Also these shells are not stiffened with either rings or axial stiffeners. The orthotropic shells are similar to the isotropic shells, however, rings or axial stiffeners or both are attached to the shell. Finally anisotropic shells are composite materials assembled of a number of layers. The material properties depend of the direction. A unified lower bound function will be derived which makes it possible to combine all test data in one chart.

2.1 Design of shells using a hand book

If one looks at the design of thin-walled shells, the ones with a major imperfection sensi-tivity, structural engineers use buckling handbooks during the design process. Typically these handbooks specify the use of the classical buckling formulas, and then multiply the load by a so-called knock-down factor, to obtain the load a shell should be able to carry. This method is an empirical approach based on historical test data. Measured buckling load data are reported by normalizing them with predictions giving the knock-down factor associated with imperfections, i.e. the fraction of the classical buckling load prediction. The experimental buckling loads are plotted with respect to the radius to thickness ratio (R/t) in Figure 2.1. On the horizontal axis the shell-wall slenderness R/t is used since the buckling stress of unstiffened shells increases linearly with t and decreases linearly with R. On the vertical axis the normalized buckling load λ, which is defined as the ratio of the experimental buckling load to the classical buckling load. The classical or theoretical

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10 Lower Bound Design Philosophy

buckling load of a thin-walled cylindrical shell [33] is Pcl = σcl2πR t = q E 3(1 − ν2₎ 2π t 2 _(2.1) where σcl = q E 3(1 − ν2₎ t R (2.2)

and E is the modulus of elasticity, and ν Poisson’s ratio. As can be noticed from Fig-ure 2.1 the knock-down factor decreases with increasing R/t. The curved solid line in the figure is the lower bound curve from which the knock-down factor is determined. It provides a good lower bound for most of the test data [34]. The data in the plot show a large scatter. If the buckling loads of the shells with R/t = 800 tested by Weingarten et al. [35] can be considered to show a normal distribution, it will be possible that new shells produced using the same manufacturing process will yield a normalized buckling load lower than the lower bound value.

On the other hand, some test results are grouped at a large distance of the lower bound curve. If one still uses the corresponding knock-down factor for these type of shells, the structure would be very safe, and therefore much too heavy. For shells to be used as rocket parts this is an argument to improve the knock-down factor. The question to answer is why do these shells perform much better than others in the plot?

Several different analytical expressions including so-called knock-down factors to be used in the design process of thin-walled shells are available. In the following part they will be discussed for different types of shells, starting with isotropic shells, continuing with stiffened isotropic shells and finally anisotropic shells.

2.2 Isotropic shells

The value calculated from the classical buckling load formula in the previous section is a theoretical value in that sense that in real life a shell will collapse at a much lower load. Sometimes the critical stress is calculated using

σc = 0.3E t

R (2.3)

which yields a buckling load of about 50% of the theoretical value as in Eq. (2.2) . The 50% is a knock-down factor on the theoretical load independent of the R/t ratio of the shell. Kanemitsu and Nojima [38] proposed the following equation:

σc = 9 E t R 1.6 + 0.16 E t L 1.2 (2.4) This equation can also be written as

σc = 9 E t R 1.6 + 0.16 E t R 1.2R L 1.2 (2.5)

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2.2 Isotropic shells 11 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 L o w er b o u n d p lo t A rb o cz & B ab co ck [5 ] B al le rs te d t & W ag n er [1 4 ] C ra te , L o & S ch w ar tz [1 6 ] D an cy & Ja co b s [2 3 ] E ss li n g er [3 6 ] H ar ri s, S u er , S k en e & B en ja m in [1 7 ] L u n d q u is t [1 3 ] R o b er ts o n [1 2 ] W ei n g ar te n , S ei d e & M o rg an [3 5 ] L o w er b o u n d , E q . (2 .7 ) λ = Pexp/Pcl R /t

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12 Lower Bound Design Philosophy 0 50 100 150 200 250 300 350 400 450 500 500 1000 1500 2000 2500 σc [M P a] R/t Eq. (2.3) SP8007 KanemitsuL/R = 2 KanemitsuL/R = 1 KanemitsuL/R = 1/2

Figure 2.2: Comparing different analytical knock-down functions

This allows the function can be plotted for different L/R ratio’s as shown in Figure 2.2. The buckling stress depends on both t/R and R/L. Compared to Eq. (2.3) the knock-down factor forR/t < 300 is much higher, yielding a larger allowable load.

Although an update is being working on, the shell design handbook still used by NASA is the well known SP-8007 report [1]. According to this report the buckling stress for isotropic shells is calculated by

σc = γ σcl (2.6)

where the knock-down factorγ is defined as γ = 1 − 0.901(1 − e−_φ ) (2.7) and φ = 1 16 s R t

To determine this formula, the test results were lumped without regard to production manufacturing methods or the method of testing. The formula can be used up to a R/t ratio of1500. Further one should be careful using this formula if L/R exceeds 5, since no experimental data of these types of shells were used to determine the empirical formula. Notice further that in Eq. (2.6) the knock-down value γ was multiplied by the classical buckling stress as shown in Eq. (2.2). The latter formula is valid for simply supported boundary conditions. As the difference between rigorous solutions are obscured by the effect of initial imperfections, this formula is used. Eqs. (2.3) and (2.6) are also plotted

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2.2 Isotropic shells 13 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 800 900 1000 λ = Pex p /P cl R/t ECCS SP8007

Figure 2.3: NASA SP8007 and ECCS lower bound formulas are nearly the same

in Figure 2.2. The figure shows that the knock-down formula from SP-8007 yields the largest buckling loads.

In Europe the commonly used handbook of the European Convention for Construc-tional Steelwork, ECCS [39] uses a similar equation1_{. This knock-down factor is a}

com-bination of two equations: γ = q 0.83 1 + 0.01R t forR/t < 212 (2.8) γ = q 0.70 0.1 + 0.01R t forR/t > 212 (2.9)

These formulas are valid for cylinders that do not exceed the limit L

R ≤ 0.95

s

R

t (2.10)

This limit is imposed to preclude the possibility of overall Euler-like column buckling interacting with shell buckling.

Comparing both definitions of the knock-down factors only shows a minor difference as shown in Figure 2.3, which is quite obvious since both factors originate from work done by Weingarten et al. [34]. However, there is a major difference between the two handbooks in the recommended procedure to be used. Whereas in using the procedure implemented in SP-8007 the use of the knock-down formula already ends the buckling

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14 Lower Bound Design Philosophy lr lr lr w¯ ¯ w ¯ w t t t Figure 2.4: Imperfections

calculation, in the ECCS handbook the quality of the shell is taken into account. The im-perfections of the shell are measured in a relatively crude manner. Using either a straight rod or a circular template the imperfections should be checked everywhere on the surface. This is shown schematically in Figure 2.4. The length of the rod or template is related to the size of the potential buckles. The ECCS proposal states that the length of the rod should be taken as

lr= 4 √

Rt (2.11)

The rod should be held anywhere against the meridian. When the ratio of the largest measured amplitude w to the corresponding l¯ r does not exceed 0.01, the knock-down factor γ given in Eqs. (2.8) and (2.9) should be used. If lr equals to 0.02, the values of γ are halved. When the ratio is in the interval 0.01 − 0.02, linear interpolation between γ and γ/2 provides the knock-down factor to be applied. For values larger than 0.02 no recommendations are given, however it seems logical a shell with such imperfections should be disposed of.

Another major difference between SP8007 and the ECCS is the recommendations of ECCS of using an extra safety factor of 4/3 for axial compressed shells, on top of the standard F.S. = 1.5. Thus one can conclude ECCS is much more conservative than SP8007.

Example: shell IW1-20

Shell IW1-20 is one of the over 30 beer cans investigated by Dancy and Jacobs [23]. The thin-walled shell manufactured from steel has a length of 100 [mm], a radius of 33 [mm] and a thickness of approximately0.1 [mm] yielding

R/t = 330. and φ = 1 16

s

R

t = 1.13537 (2.12)

Then the lower bound value is γ = 1 − 0.901(1 − e−φ

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2.3 Orthotropic shells 15 R t y z d1 c1 es ds R t x z d2 c2 er dr

Stringer stiffened Ring stiffened

Figure 2.5: Orthotropic shell The lower bound buckling load for this shell now becomes

Pγ = γ.Pcl = −3102.4 N (2.14)

Comparing this to the experimentally found load

Pexp = −3890.0 N (2.15)

one notices the lower bound value is conservative. Looking at all the buckling load data of the beer cans of Dancy and Jacobs [23] as plotted in Figure 2.1 it can be seen that these values show a large spread, where the minimum buckling load is positioned on the lower bound curve. Therefore, for the beer cans there is no gain in spending energy in the improvement of the lower bound curve.

It is interesting to mention that in the design of the beer cans other requirements ex-ist which determine the wall thickness of the cans. In decreasing the wall thickness any further the can might be damaged by sharp finger nails. On the other hand, can manu-facturers are interested in the loading capability of shells where certain imperfections are put on the shell surface on purpose. Think of, for example, embossed company logos.

2.3 Orthotropic shells

Shells can be stiffened using axial stiffeners, rings or a combination of both. Let us con-sider a thin-walled cylindrical shell, reinforced by closely spaced circular rings attached on the outside of the shell and with longitudinal stringers attached on the inside, as il-lustrated in Figure 2.5. If the stiffener spacing is small enough the stiffener effects are smeared over the shell. Whether or not this yields satisfying results depends on the buck-ling mode. As a rule of thumb at least 5 stringers or rings should be situated on one half

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wave of the buckling mode. If there are less, a discrete stiffener theory should be used instead since the smeared stiffener wall assumptions become invalid [40]. For the latter theory the shells will be referred to as stiffened isotropic shells.

For the smeared theory the cylinder is approximated by a fictitious sheet whose orthotropic bending and extensional properties include those of the individual stiffening elements av-eraged out over representative widths or areas. The smeared bending stiffness per unit width of the wall ¯Dx and ¯Dy in _{x− and y− direction respectively, and the smeared} ex-tensional stiffness’s of the wall ¯Exand ¯Ey in_{x− and y− direction respectively are} repre-sented by ¯ Dx = D(1 + η01), Ex¯ = C(1 + µ1) (2.16) ¯ Dy = D(1 + η02), Ey¯ = C(1 + µ2) (2.17) in which D = Et 3 12(1 − ν2₎, C = Et 1 − ν2

The terms, which are a function of the dimensions of the stringers and rings, are: η01 = EI01

dsD and I01 = Is+ Ase 2

s (2.18)

η02 = EI02

drD and I02 = Ir+ Are 2 r (2.19) µ1 = (1 − ν2) As dst (2.20) µ2 _{= (1 − ν}2)Ar drt (2.21)

wheredsanddrare the stringer and ring spacing respectively, andesanderthe eccentric-ity of the stringer and the ring. The areas and the area moments of inertia of the stringers are As= c1d1 (2.22) Is = 1 12c1d 3 1 (2.23)

Similarly for the rings:

Ar= c2d2 (2.24) Ir = 1 12c2d 3 2 (2.25)

The contribution of the stringers and rings of the shell in the change of the critical buckling load can be implemented in the knock-down formula by modifying the wall thickness of

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2.3 Orthotropic shells 17

the shell. For orthotropic shells this lower bound formula is similar to the one for isotropic shells: γs_{= 1 − 0.901(1 − e}−φ_s ) (2.26) where φs= 1 29.8 s R t∗ and t ∗ = 4 v u u t ¯ DxDy¯ ¯ ExEy¯ where the adjusted wall thickness t∗

is a function of the bending stiffness and the exten-sional stiffness of the stiffened shell. Whereas for isotropic shells the knock-down factor was a function of R/t, for orthotropic shells it also depends on the number and size of stiffeners which are implemented in the adjusted thicknesst∗

.

The lower bound curve for the orthotropic shells is plotted together with experimental data in Figure 2.6. On the vertical axis there is a difference noticeable when comparing it to Figure 2.1. In Figure 2.6 ρ is the normalized load, where as a normalization term the theoretical buckling load of a orthotropic shell is used. Thus

ρ = Pexp

Pstf (2.27)

where

Pstf = λmCkℓPcl Here λm

Ckℓ is the critical (lowest) eigenvalue of the linearized stability equations using membrane prebuckling [41]: λm Ckℓ= 1 2 ( ¯ γD,k,ℓ α2 k +(¯γQ,k,ℓ+ α 2 k) 2 α2 kγH,k,ℓ¯ ) (2.28) where ¯ γD,k,ℓ = Dxxα¯ 4_k+ ¯Dxyα2_kβ_ℓ2+ ¯Dyyβ_ℓ4 , α2_k = k2Rt 2c π L 2 ¯ γH,k,ℓ = Hxxα¯ 4_k+ ¯Hxyα2_kβ_ℓ2+ ¯Hyyβ_ℓ4 (2.29) ¯ γQ,k,ℓ = Qxxα¯ 4_k+ ¯Qxyα2_kβ_ℓ2+ ¯Qyyβ_ℓ4 , β_ℓ2 = ℓ2Rt 2c 1 R 2

andPclis the classical buckling load of a thin-walled cylindrical shell defined in Eq. (2.1). The stiffness parameters and the wave number parameters have been defined in Ap-pendix B. An interesting fact is the value of λmCkℓ for isotropic shells, since for those shellsλm

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18 Lower Bound Design Philosophy 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 2000 L o w er b o u n d p lo t A R -s h el ls h ea v y [5 ] A R -s h el ls li g h t [5 ] A R -s h el ls m ed iu m [5 ] A S -s h el ls [5 ] R O st ri n g er s [2 1 ] A B -s h el ls [2 2 ] K R -s h el ls [2 2 ] L o w er b o u n d st if fe n ed is o tr o p ic , E q . (2 .2 6 ) ρ = Pexp/(λmCkℓPcl) R /t ∗

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2.4 Anisotropic shells 19

Example: shell AS 2

Shell AS 2 is one of three stringer stiffened shells investigated by Arbocz and Babcock [4, 5]. The aluminium 6061-T6 shell has a length of139.7 [mm], a radius of 101.6 [mm] and a wall thickness of 0.197 [mm]. Taken into account the properties of the 80 axial stringers:

As = 0.7987 [mm2_] _ds_{= 8.0239 [mm]}

Is= 0.015038 [mm4_] _es_{= 0.3368 [mm]} (2.30)

using Poisson’s ratioν = 0.3, the adjusted t∗

is calculated using Eqs. (2.16) - (2.26): t∗ = 0.1091 [mm] (2.31) yielding aR/t∗ ratio of R/t∗ = 931.5908 (2.32)

Notice this value for t∗

is smaller than the actual wall thickness because of the definition shown in Eq. (2.26). In section 2.5 a different thickness will be introduced larger than the actual wall thickness. This latter definition seems more convincing because of the expected higher buckling load compared to an unstiffened shell having the same wall thickness.

The lower bound value of the orthotropic shell is γs_{= 1 − 0.901(1 − e}−_φ_s ) = 0.4238 (2.33) using φs= 1 29.8 q R/t∗ = 1.02036

Although the value of t∗

is almost twice as low as the wall thicknesst, the knock-down factor is higher than for a shell without stiffeners using the same wall thickness, sinceφs is used in stead ofφ. The lower bound buckling load for this shell:

Pγ = γs_{.Pstf = γ}s. λm_CkℓPcl _{= −1396.0 lbs} (2.34) Comparing this to the experimentally found load

Pexp = −3211.7 lbs (2.35)

one notices the lower bound value is very conservative. This again shows an update of the knock-down parameters is needed.

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{

middle surface layer number Inner Middle Outer Layers Fibre orientation θ 1 1 2 2 k N L Generatrix x y R t z z t/2 z0 _z1 z2 zk zk−1 zN zN −1

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2.4 Anisotropic shells 21

2.4 Anisotropic shells

Anisotropic shells are typically shells constructed out of several layers of a composite material as shown in Figure 2.7. Each layer is a curved arrangement of unidirectional fibers or woven fibers in a matrix. The fibers carry almost all the load whereas the function of the matrix is to support and protect the fibers and to provide a means of distributing load among and transmitting load between the fibers. The extensional stiffness termsAij and the bending stiffnessDij are defined as

Aij = N X k=1 ( ¯Qij)k(zk− zk−1) Bij = 1 2 N X k=1 ( ¯Qij)k(zk2− zk−12 ) Dij = 1 3 N X k=1 ( ¯Qij)k(z_k3_{− z}_k−13 ) (2.36) where ¯

Q11 = Q11cos4θ + 2(Q12+ 2Q66) sin2θ cos2θ + Q22sin4θ ¯

Q12 = (Q11+ Q22− 4Q66) sin2θ cos2θ + Q12(sin4θ + cos4θ) ¯

Q22 = Q11sin4θ + 2(Q12+ 2Q66) sin2θ cos2θ + Q22cos4θ ¯

Q16 = (Q11_{− Q}22− 2Q66) sin θ cos3θ + (Q12− Q22+ 2Q66) sin3θ cos θ ¯

Q26 = (Q11_{− Q}22− 2Q66) sin3θ cos θ + (Q12− Q22+ 2Q66) sin θ cos3θ ¯

Q66 = (Q11+ Q22− 2Q12− 2Q66) sin2θ cos2θ + Q66(sin4θ + cos4θ) (2.37) and the reduced stiffness

Q11 = E1 1 − ν12ν21 Q12 = ν12E2 1 − ν12ν21 = ν21E1 1 − ν12ν21 Q22 = E2 1 − ν12ν21 Q66 = G12 (2.38)

according to Jones [42]. HereE1andE2are the Young’s moduli in the1 and 2 directions, respectively, and νij is the Poisson’s ratio for transverse strain in the j-direction when stressed in thei-direction. Further, G12is the shear modulus in the_{1 − 2 plane.}

For anisotropic shells the lower bound formula is chosen similar to those of the isotropic shells and orthotropic shells:

γa_{= 1 − 0.901(1 − e}−_φ_a ) (2.39) where φa= 1 29.8 s R t+ (2.40)

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22 Lower Bound Design Philosophy and t+= 4 s D11D22 A11A22 (2.41)

andt+_{is the adjusted wall thickness for anisotropic shells. Notice that in the formula for} t+_{the extensional stiffness terms and the bending stiffness terms are used as before in the} definition oft∗

in Eq. (2.26).

The lower bound curve for the anisotropic shells is plotted in Figure 2.8 together with experimental data. Similar to the plot with the experimental results for the orthotropic shells, the value ofR/t+_{is used on the horizontal axis, where}_t+_{is the adjusted thickness} of the shell, which includes the effect of the composite material. On the vertical axis one can find the non-dimensional parameterρ, the normalized load. As a normalization term the theoretical buckling load of an anisotropic shell is used. Thus

ρ = Pexp

Pani (2.42)

where

Pani = λmnτPcl

Furtherλmnτ is the critical (lowest) eigenvalue of the linearized stability equations using membrane prebuckling of the anisotropic shell [31]:

λmnτ = 1 2α2 m+ αp2 T1,m,n¯ + ¯T2,p,n+ ¯ T2 3,m,n ¯ T2 5,m,n + T¯ 2 4,p,n ¯ T2 6,p,n ! (2.43) where ¯ T1,m,n = ¯γe D∗_,m,n− ¯γ_Do∗_,m,n T4,p,n¯ = ¯γ_Be∗_,p,n+ ¯γo_B∗_,p,n+ α2_p ¯ T2,p,n= ¯γe D∗_,p,n+ ¯γ_Do∗_,p,n T5,m,n¯ = ¯γ_Ae∗_,m,n+ ¯γ_Ao∗_,m,n ¯ T3,m,n = ¯γe B∗_,m,n− ¯γ_Bo∗_,m,n+ α2_m T6,p,n¯ = ¯γ_Ae∗_,p,n− ¯γ_Ao∗_,p,n (2.44) The coefficientsγ¯e A∗_,m,n,¯γ_Ao∗_,m,n,¯γ_Ae∗_,p,n,¯γ_Ao∗_,p,n,¯γ_Be∗_,m,n,¯γ_Bo∗_,m,n,γ¯_Be∗_,p,n,γ¯_Bo∗_,p,n,¯γ_De∗_,m,n, ¯ γo

D∗_,m,n, γ¯_De∗_,p,n and γ¯o_D∗_,p,n are functions of the stiffness parameters Aij, Bij and Dij.

Their definitions can be found in Appendix B. Both terms α2

m and α2p are functions of the geometry of the shell and the number of waves of the buckling mode, also defined in Appendix B. Notice that the eigenvalueλmnτ depends on the wave numbersm and n and on Khot’s skewedness parameter τK [43]. This skewedness parameter is introduced in order to account for the possibility of bending-twisting coupling.

Example: shell AW-CYL-1-1

Shell AW-CYL-1-1 is one of five layered, composite graphite-epoxy cylinders investi-gated by Waters [25]. The shell has a length of 14 [in], a radius of 7.99945 [in] and a thickness of _{0.039976 [in]. The shell has gotten 8 layers, lay-up [±45/0/90]}s, each ply

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2.4 Anisotropic shells 23 0 0.2 0.4 0.6 0.8 1 1.2 0 500 1000 1500 2000 L o w er b o u n d p lo t H ¨u h n e [3 7 ] W at er s [2 5 ] L o w er b o u n d an is o tr o p ic , E q . (2 .3 9 ) ρ = Pexp/(λmnτPcl) R /t +

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has a thickness of0.004997 [in]. Then the stiffness terms in Eq. (2.36) can be calculated as

A11= 0.326879.106 _D11_{= 39.8701}

A22= 0.326879.106 _D22_{= 31.3826} (2.45)

Taken into account these properties the adjustedt+ _{is calculated using Eq. (2.41) as:}

t+= 0.0104026 [in] (2.46)

yielding aR/t+_{ratio of}

R/t+ = 768.9850 (2.47)

Then the lower bound value is γa_{= 1 − 0.901(1 − e}−_φ_a ) = 0.35530 (2.48) using φa= 1 29.8 q R/t+ _{= 0.93056}

The lower bound buckling load for this shell:

Pγ = γa_{.Pani = γ}a.λmnτPcl _{= −14629 lbs} (2.49)

Comparing this to the experimentally found load

Pexp = −30164 lbs (2.50)

Notice the lower bound value is very conservative for this shell. The lower bound for the other 4 shells in the group withR/t+ about 400 is also conservative, however not a lot of weight can be saved on these shell if the bound is improved, Figure 2.8.

2.5 Unified lower bound curve

In the NASA report SP8007 [1] the functions for the lower bound curves for isotropic and orthotropic shells are not the same. The lower bound curve for the isotropic shell is a function ofR/t, whereas for the orthotropic shell it is a function of R/t∗

, wheret∗ has been defined as:

t∗ = 4 v u u t ¯ DxDy¯ ¯ ExE¯y (2.51) This definition has a disadvantage. If the stringer and ring areas of the shell are very small, and the shell can be considered to act as an isotropic shell, the parameters η01,η02, µ1 andµ2defined in Eqs. (2.18)-(2.21) will go to zero. The stiffness terms simplify to

¯

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2.5 Unified lower bound curve 25 ¯ Dy _{⇒ D} ¯ Ex ⇒ C ¯ Ey ⇒ C

Substituting this result into Eq. (2.51) yields a thickness t∗

= √t

12 (2.52)

Using this formula fort∗

in the lower bound formula of the orthotropic shell should yield the equation of the isotropic shell, therefore in Eq. (2.26) a coefficient 1/29.8 in the ex-pression forφsis used in stead of1/16 in φ in Eq. (2.7).

Although the definition of t∗

in Eq. (2.51) is an elegant formula, it is reasonable that in the limit where the stiffeners are negligibly small, the adjusted thickness is

t∗ = t

In stead of the definition in Eq. (2.51) it is suggested to multiply this formula by √12 yielding tu =√12 t∗ =√12 4 v u u t ¯ DxDy¯ ¯ ExE¯y (2.53) which can be rewritten using Eqs. (2.16) and (2.17) to

tu = 4 v u u t (1 + η01) + (1 + η02) (1 + µ1) + (1 + µ2) t (2.54)

A benefit is that the lower bound formula Eq. (2.7) can also be used for the stiffened shell. Similar to the stiffened shell, for anisotropic shells another definition for t+ _{as an} alternative to Eq. (2.41) will be used. Let

tu =√12 t+ =√12 4

s

D11D22

A11A22 (2.55)

which is in fact equal to the definition as used in Eq. (2.41) multiplied by √12. For a single layer of isotropic material with material properties E and ν and thickness t the stiffness terms are

¯ Q11 = Q11 ¯ Q22 = Q22 and Q11 = Q22 = E 1 − ν2 (2.56)

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26 Lower Bound Design Philosophy then A11 = Q11(z1− z0) = Q11(t/2 − (−t/2)) = Et 1 − ν2 A22 = Q22(z1− z0) = Q22(t/2 − (−t/2)) = Et 1 − ν2 D11 = 1/3 Q11(z31 − z30) = 1/3 Q11 (t/2)3− (−t/2)3= Et 3 12(1 − ν2₎ D22 = 1/3 Q22(z3₁ − z30) = 1/3 Q22 (t/2)3− (−t/2)3= Et 3 12(1 − ν2₎

Substituting these equations into the formula for the adjusted wall thickness, Eq. (2.55):

tu = t (2.57)

Also for the anisotropic shells the lower bound formula Eq. (2.7) can be used in the design of a new shell.

Recall shell AS 2 in the example on page 19 where the wall thickness t∗

is almost half the actual wall thickness. Calculation of the unified thickness yields

t =√_{12 × 0.1091 = 0.3779 [mm]} (2.58)

which is about twice the value of the wall thickness of the wall. This result seems more appropriate.

In the following examples the unified lower bound theorem can be used to generate one plot containing the experimental results of shells where the walls consists of isotropic material, possibly stiffened with axial stringers and/or rings, and walls built up using a set of anisotropic layers.

Example: comparison of stiffened and unstiffened shells

The shells reported in [5] can now be plotted in a single lower bound plot, which makes it easier to compare different types of shells. The shells in the report are the results of imperfection surveys carried out at Caltech in the sixties of the last century. The shells consist of a set of copper electroplated shells (A-shells), nickel electroplated shells (N-shells), machined brass shells (B-(N-shells), welded stainless steel shells (ST-(N-shells), stringer stiffened aluminium shells (AS-shells), and finally ring stiffened aluminium shells (AR-shells). The results are plotted in Figure 2.9. Notice on the vertical axis the in the normal-ization the factor λm

u has been used. For orthotropic shells this will be replaced byλmCkℓ, for anisotropic shells byλmnτ.

The first thing which can be observed is the clustering of the data. The shells within a set, which have about the sameR/t ratio, show a small spread of the buckling load. This is important as this makes it realistic to move the lower bound line upwards.

However, the ST-shells all collapsed before the load associated with the lower bound curve was reached. These shells were reported to have shown plastic buckling [5]. The

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2.5 Unified lower bound curve 27 0 0.2 0.4 0.6 0.8 1 1.2 0 200 400 600 800 1000 1200 L o w er b o u n d p lo t ρ = Pexp/(λm uPcl) R /t u L o w er b o u n d , E q . (2 .7 ) A -s h el ls A R -s h el ls h ea v y A R -s h el ls li g h t A R -s h el ls m ed iu m A S -s h el ls B -s h el ls N -s h el ls S T -s h el ls

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shells were cut from commercial, longitudinal welded, type 304 stainless steel tubing. As the fabrication process cannot be compared with the carefully procedure as followed for the other shells, and the buckling behaviour is completely different, they will not be discussed any further.

The buckling loads in Figure 2.9 are all normalized. Recall that for isotropic shells the normalization factor

λm_Ckℓ= 1.0

The normalized buckling load for the isotropic unstiffened shells is lower than for the stiff-ened orthotropic shells. The normalization factor λm

Ckℓ is calculated for a stiffened shell with simply supported SS-3 boundary condition but the ends of the shells are clamped during the test, approaching CC-4. Because of this clamped boundary condition the cal-culated buckling load will be lower than should be expected, yielding a conservative re-sult [40].

Notice the buckling load of the isotropic A-shells, B-shells and N-shells seems to be in-dependent of the R/t ratio, despite of the definition of the knock-down factor. No real conclusion can be drawn because of the small number of tests within these sets.

2.6 Discussions and conclusion

In this chapter lower bound plots have been shown for isotropic shells, orthotropic shells and anisotropic shells. The lower bound curves plotted in these figures are empirical equations used as a knock-down value to determine a safe buckling load of a shell design. Notice that the plots are similar: the empirical functions for the lower bound curves of the isotropic shells, orthotropic shells and anisotropic shells are the same if the stiffness parameters valid for isotropic shells are substituted in each of the three equations (2.7), (2.26) and (2.39).

The lower bound curves originate from the NASA report SP-8007 [1] and are based on experimental data dated before 1968. More recent data have been plotted into these figures. These newer shells manufactured using newer production techniques will be de-signed much more conservative than shells dede-signed many years ago, because the lower bound curve has not been adjusted for modern technology where the quality and repeata-bility have been improved. Using the ECCS handbook [39], which is also based on the experimental data used for SP-8007, is even more conservative since an extra safety factor is recommended for axial compressed shells. Substantial weight savings can be achieved if the lower bound theory is improved, which is possible if the effect of modern technol-ogy is used in the theory.

The lower bound formula for isotropic shells is used up to aR/t ratio of 2500. Notice that the formula which can be used up to 1500 according to SP-8007 also yields smaller buckling loads for the higherR/t values as shown in Figure 2.1!

The experimental buckling loads found for the beer cans show a large spread. Since the lowest buckling load is found on the lower bound curve, there is no profit in spending energy in the improvement of the lower bound curve for these cans.

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2.6 Discussions and conclusion 29

The normalized buckling loads of some shells are higher than 1.0 in the lower bound plot for the orthotropic shells in Figure 2.6. Because the buckling loads are normalized using the theoretical buckling load of a orthotropic shell with simply supported boundary conditions, this can be expected for shells tested with clamped boundary conditions.

One should stay compatible with the lower bound curves defined in SP-8007 [1], a handbook of which the curves have been the basis for newer hand books as well. There-fore the lower bound curve for the anisotropic shells has been based on the curve for the orthotropic shells. At the end of this chapter a unified lower bound curve has been pro-posed which has the advantage that all data can be plotted in one figure. Furthermore, in this new approach the adjusted thickness will be equal to the real thickness in the limit case where the areas of the stringers and rings go to zero.

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The imperfection data bank and its applications

The Imperfection Data Bank and its

Applications

Proefschrift

Contents

Nomenclature

Abstract

Samenvatting

Chapter 1

Introduction

1.1

The shell design procedure

1.2

Why are imperfections important for the design of

shells?

1.3

Building an imperfection data bank

1.4

Layout of the thesis

Chapter 2

Lower Bound Design Philosophy

2.1

Design of shells using a hand book

2.2

Isotropic shells

Example: shell IW1-20

2.3

Orthotropic shells

Example: shell AS 2

{

2.4

Anisotropic shells

Example: shell AW-CYL-1-1

2.5

Unified lower bound curve

Example: comparison of stiffened and unstiffened shells

2.6

Discussions and conclusion